Consider the GARCH(1,1)model $\sigma _ { t } ^ { 2 }$ = ?₀ + ?₁
$u _ { t - 1 } ^ { 2 }$ + ?₁
$\sigma _ { t - 1 } ^ { 2 }$ Show that this model can be rewritten as $\sigma _ { t } ^ { 2 }$ = $\frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } }$ + ?₁( $u _ { t - 1 } ^ { 2 }$ + ?₁
$u _ { t - 2 } ^ { 2 }$ + $\phi _ { 1 } ^ { 2 }$ $u _ { t - 3 } ^ { 2 }$ + $\phi _ { 1 } ^ { 3 }$ $u _ { t - 4 } ^ { 2 }$ + ...). (Hint: use the GARCH(1,1)model but specify it for $\phi _ { t - 1 } ^ { 2 }$ ; substitute this expression into the original specification, and so on.)Explain intuitively the meaning of the resulting formulation.

Question

Consider the GARCH(1,1)model

\sigma _ { t } ^ { 2 }

= ?₀ + ?₁

u _ { t - 1 } ^ { 2 }

+ ?₁

\sigma _ { t - 1 } ^ { 2 }

Show that this model can be rewritten as

\sigma _ { t } ^ { 2 }

=

\frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } }

+ ?₁(

u _ { t - 1 } ^ { 2 }

+ ?₁

u _ { t - 2 } ^ { 2 }

+

\phi _ { 1 } ^ { 2 }

u _ { t - 3 } ^ { 2 }

+

\phi _ { 1 } ^ { 3 }

u _ { t - 4 } ^ { 2 }

+ ...). (Hint: use the GARCH(1,1)model but specify it for

\phi _ { t - 1 } ^ { 2 }

; substitute this expression into the original specification, and so on.)Explain intuitively the meaning of the resulting formulation.

Quizplus · Accepted Answer

The Answer of Consider the GARCH(1,1)model $\sigma _ { t } ^ { 2 }$ = ?₀ + ?₁ $u _ { t - 1 } ^ { 2 }$ + ?₁ $\sigma _ { t - 1 } ^ { 2 }$ Show that this model can be rewritten as $\sigma _ { t } ^ { 2 }$ = $\frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } }$ + ?₁( $u _ { t - 1 } ^ { 2 }$ + ?₁ $u _ { t - 2 } ^ { 2 }$ + $\phi _ { 1 } ^ { 2 }$ $u _ { t - 3 } ^ { 2 }$ + $\phi _ { 1 } ^ { 3 }$ $u _ { t - 4 } ^ { 2 }$ + ...). (Hint: use the GARCH(1,1)model but specify it for $\phi _ { t - 1 } ^ { 2 }$ ; substitute this expression into the original specification, and so on.)Explain intuitively the meaning of the resulting formulation.

Consider the GARCH(1,1)model σt2\sigma _ { t } ^ { 2 }σt2​

Consider the GARCH(1,1)model $\sigma _ { t } ^ { 2 }$